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An Invitation to Model-Theoretic Galois Theory

Published online by Cambridge University Press:  15 January 2014

Alice Medvedev  and
Ramin Takloo-Bighash
Alice Medvedev
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinoisat Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USAE-mail:, alice@math.uic.edu, E-mail:, rtakloo@math.uic.edu
Ramin Takloo-Bighash
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinoisat Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USAE-mail:, alice@math.uic.edu, E-mail:, rtakloo@math.uic.edu
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Abstract

We carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 2 , June 2010 , pp. 261 - 269
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1] Artin, Emil, Galois Theory, Notre Dame Mathematical Lectures, no. 2, University of Notre Dame, Notre Dame, Ind., 1942, edited and supplemented with a section on applications by Arthur N. Milgram.Google Scholar
[2] Chase, S. U., Harrison, D. K., and Rosenberg, Alex, Galois theory and Galois cohomology of commutative rings, Memoirs of the American Mathematical Society, vol. 52 (1965), p. 15–33.Google Scholar
[3] Evans, David M. and Pastori, Elisabetta, Second cohomology groups and finite covers, 2009.Google Scholar
[4] Giraud, Jean, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[5] Harris, Michael and Taylor, Richard, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, with an appendix by Vladimir G. Berkovich.Google Scholar
[6] Krull, Wolfgang, Galoissche Theorie der unendlichen algebraischen Erweiterungen, Mathematische Annalen, vol. 100 (1928), no. 1, pp. 687–698.CrossRefGoogle Scholar
[7] Neukirch, Jürgen, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder.CrossRefGoogle Scholar
[8] Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996.Google Scholar
[9] Pillay, Anand, Remarks onGalois cohomology and definability, The Journal of Symbolic Logic, vol. 62 (1997), no. 2, pp. 487–492.Google Scholar
[10] Poizat, Bruno, Une théorie de Galois imaginaire, The Journal of Symbolic Logic, vol. 48 (1983), no. 4, pp. 1151–1170 (1984).Google Scholar
[11] Poizat, Bruno, A course in model theory, Universitext, Springer-Verlag, New York, 2000, translated from the French by Klein, Moses and revised by the author.CrossRefGoogle Scholar
[12] Poizat, Bruno, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001, translated from the 1987 French original by Moses Gabriel Klein.Google Scholar
[13] Rasala, Richard, Inseparable splitting theory, Transactions of the American Mathematical Society, vol. 162 (1971), pp. 411–448.CrossRefGoogle Scholar
[14] Satake, I., Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics, vol. 3, Marcel Dekker Inc., New York, 1971, with an appendix by M. Sugiura, notes prepared by Doris Schattschneider.Google Scholar
[15] Shelah, Saharon, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1978.Google Scholar
[16] Steinitz, Ernst, Algebraische Theorie der Körper, Chelsea Publishing Co., New York, N. Y., 1950.Google Scholar
[17] Weber, H., Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie, Mathematische Annalen, vol. 43 (1893), no. 4, pp. 521–549.Google Scholar
[18] Weil, André, Foundations of algebraic geometry, American Mathematical Society Colloquium Publications, vol. 29, American Mathematical Society, New York, 1946.Google Scholar
[19] Wiles, Andrew, Modular elliptic curves and Fermat's last theorem, Annals of Mathematics. Second Series , vol. 141 (1995), no. 3, pp. 443–551.Google Scholar